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Metamath Proof Explorer

Theorem equsb3r

Description: Substitution applied to the atomic wff with equality. Variant of equsb3 . (Contributed by AV, 29-Jul-2023) (Proof shortened by Wolf Lammen, 2-Sep-2023)

		Ref	Expression
	Assertion	equsb3r	⊢ ( [ 𝑦 / 𝑥 ] 𝑧 = 𝑥 ↔ 𝑧 = 𝑦 )

Proof

Step	Hyp	Ref	Expression
1		equequ2	⊢ ( 𝑥 = 𝑤 → ( 𝑧 = 𝑥 ↔ 𝑧 = 𝑤 ) )
2		equequ2	⊢ ( 𝑤 = 𝑦 → ( 𝑧 = 𝑤 ↔ 𝑧 = 𝑦 ) )
3	1 2	sbievw2	⊢ ( [ 𝑦 / 𝑥 ] 𝑧 = 𝑥 ↔ 𝑧 = 𝑦 )